It is well-known to those familiar with the art of exploration seismology that the rocks of the earth's subsurface exhibit seismic anisotropy, and that failure to account for this can confuse the interpretation of seismic data. For example, surface-reflection seismic data acquired by experiments in anisotropic rock, but interpreted via isotropic theory, can yield incorrect estimates of reflection depths, incorrect analysis of Amplitude-vs.-Offset (AVO) measurements, etc.
Seismic anisotropy is that property of materials by which the velocity of seismic waves (taken here as P-waves) depends upon the direction of travel of such waves. Polar anisotropy is that special case where the variation is primarily or exclusively with the polar angle (i.e., the angle from the vertical, rather than the azimuthal angle). It can be caused, for example by preferred alignment of rock grains (as in shales), or by sedimentary layering on a scale much smaller than the seismic wavelength. Because such layering is common in the earth's crust, and because seismic waves of differing wavelength are affected differently by such layering, the anisotropy measured by any seismic experiment depends upon the wavelengths used in the experiment Hence, experimental results from widely differing wavelength bands may not be combined to estimate the anisotropy (more on this later). Since subsurface rocks in different places possess seismic anisotropy to varying degrees, accurate local measurements of the anisotropy are routinely required in order to make corrections, thereby avoiding errors such as those mentioned above.
FIG. 1 is a good starting place to illustrate the problem. In a typical seismic experiment, seismic waves are emitted by a source at S, reflected by a rock-layer at depth, z and received by a receiver at R, separated from S by a distance x. With the geometry shown, and if the rock is uniform and isotropic, it is well-known that the arrival time t(x) of the wave arriving at offset x is given by:                                           t            2                    ⁡                      (            x            )                          =                              t            0            2                    +                                    x              2                                      V              2                                                          (        1        )            where t0 is the vertical travel time or the arrival time (of the wave) at the special place where x=0 (i.e., the place of normal incidence), and where V is the velocity of seismic waves. The quantities V and t0 are conventionally determined by measurements of t(x), with V determined as the parameter which best fits the hyperbolic variation of arrival time with offset The (initially unknown) depth z to the reflector is given by:                     z        =                              V            ⁢                                                   ⁢                          t              0                                2                                    (        2        )            With V and t0 measured, the depth z is thereby determined.
Of course, in any practical situation, the geometry will be more complicated than that shown, and the material properties will not be uniform. For example, FIG. 2 illustrates the situation where the reflector is dipping at angle φ. It is well known (See Slotkin, M. M., “Lessons in Seismic Computing,” Soc. Expl. Geoph., 1959, Tulsa, Okla., the teachings of which are incorporated herein by reference) that:                                           t            2                    ⁡                      (            x            )                          =                              t            0            2                    +                                    x              2                                      V              2                                +                                    (                                                2                  ⁢                                      t                    0                                    ⁢                  x                                V                            )                        ⁢            sin            ⁢                                                   ⁢            ϕ                                              (        3        )            where t0 is given by Equation (2). However, the simple situation shown in FIG. 1 suffices to illustrate the difficulties which arise if the rock is considered to be anisotropic.
If the rock is anisotropic, then Equation (1) no longer applies as written. To generalize it for anisotropy, one has to specify what sort of anisotropy the rocks possess. In the simplest anisotropic case, the velocity varies only with the incidence angle θ shown in FIG. 1, and not with azimuthal angle. In this (i.e., polar anisotropic) case, it is well-known that the angular variation of velocity depends upon four material parameters, following a complicated formula known to those skilled in the art.
However, it is also well-known that, if the anisotropy is “weak,” then the P-wave velocity Vp(θ) depends upon only three parameters which may be chosen (modifying Thomsen, “Weak elastic anisotropy,” Geophysics, 1986, 51(10), pp. 1954-1966, the teachings of which are incorporated herein by reference) as:VP(θ)=V0[1+δ sin 2(θ)+η sin 4(θ)]  (4) where V0 is the velocity in the special case that θ=0 (i.e. it is the vertical velocity), and δ and η are two non-dimensional anisotropic parameters. Materials with “weak” polar anisotropy have both δ and η much less than one. The horizontal velocity Vp(η=90″) is given by:Vp(θ=90°)=V0[1+δ+η]  (5) so thatε=δ+η  (6) is the horizontal anisotropy parameter. The parameter δ controls the near-vertical variation of velocity, and η is called the anelliptic anisotropic parameter.
With this notation, one can generalize Equation (1) to the Vertical Traversely Isotropic case:                                           t            2                    ⁡                      (            x            )                          =                              t            0            2                    +                                    x              2                                      V              mo              2                                                          (        7        )            for small offsets X≦Z. The parameter Vmo called the “moveout velocity” or dx/dt, is given in terms of quantities previously defined by:Vmo=V0√{square root over (1+2δ)}≅V0[1+δ]  (8) Further, the unknown depth z is given, in terms of the vertical velocity V0 or dz/dt, by:                     z        =                                            V              0                        ⁢                          t              0                                2                                    (        9        )            (compare to Equation (2)). Those skilled in the art will recognize that the vertical velocity and the moveout velocity are features of the P-wave velocity function.
However, the vertical velocity V0 is not given by the hyperbolic variation of t with x, which is Vmo and which includes an anisotropic correction (see Equation (8)). Hence, although Vmo is known, V0 itself is not known, and so the measured vertical travel time t0 may not be interpreted to yield the depth. Thus, because of the presence of anisotropy, neither the anisotropy itself nor even the depth can be determined accurately.
If longer offsets, x>z, are acquired, then the arrival times are given by a formula more complicated than that given by Equation (7) (i.e., with an additional (non-hyperbolic) term involving the anellistic anisotropy parameter η; See Tsvankin and Thomsen, “Nonhyperbolic Reflection Moveout in Anisotropic Media,” Geophysics, 1994, 59(8), pp. 1290-1304, the teachings of which are incorporated herein by reference). This non-hyperbolic term has been specified (See Alkhalifah, “Anisotropy Processing in Vertically Inhomogeneous Media”, Soc. Expl. Geoph, Expdd. Absts., 1993, 65, 348-353, the teachings of which are incorporated herein by reference) in terms of η, generalized as:   η  =                    (                  ɛ          -          δ                )                    (                  1          +                      2            ⁢            δ                          )              .  However, this additional term helps determine the anellyptic anisotropy parameter η, rather than the vertical velocity V0 and/or the near-surface anisotropy parameter δ, so the problem remains.
Heretofore, the only practical seismic way to determine the vertical velocity V0 has been to measure it (e.g., by drilling a hole, and by using Equation (8)). Thus, when both the vertical velocity V0 and the moveout velocity Vmo are determined, the near-surface anisotropy parameter δ may by found using Equation (8). This set of operations then determines all three parameters (V0, δ, η) with seismic resolution (i.e., with the resolution of a seismic wavelength or more (e.g., several hundred meters)). For some purposes, this is sufficient, but for others it is not For example, proper interpretation of the AVO effect requires an estimate of the near-surface anisotropy parameter δ with much greater resolution. The AVO effect is often used to detect the presence of gas in the pore-space of the rock, but this detection can be confused by anisotropy.
The argument is as follows:
After certain “true relative amplitude” corrections have been performed, the amplitudes of the received signals in the reflection experiment of FIG. 1 may be interpreted in terms of the reflectivity. It is well-known (See Thomsen, “Weak Anisotropic Reflections, in Offset-Dependent Reflectivity—Theory and Practice of AVO Analysis,” Castagna, ed., Soc. Expl Geoph., 1993, Tulsa, pp. 103-114, the teachings of which are incorporated herein by reference) that, if the contrast in elastic properties across the reflecting horizon is small, the reflectivity at small-to-moderate angles may be written in the isotropic case as:Rp(θ)=R0+R1 sin 2(θ)  (10) where                               R          0                =                                            Δ              ⁢                                                           ⁢              Z                                      2              ⁢              Z                                ⁢                                           ⁢          and                                    (                  10          ⁢          a                )                                          R          1                =                              1            2                    ⁢                      (                                                            Δ                  ⁢                                                                           ⁢                  V                                V                            -                                                                    (                                                                  2                        ⁢                                                  V                          s                                                                                            V                        p                                                              )                                    2                                ⁢                                                      Δ                    ⁢                                                                                   ⁢                    G                                    G                                                      )                                              (                  10          ⁢          b                )            Here, Z is the average P-wave impedance across the horizon, Vs is the average shear-wave velocity, and G is the average shear modulus. ΔZ, ΔG and ΔV are the corresponding jumps across the reflecting horizon. Anomalous values of R1 result from the presence of gas on one side of the horizon, and this is a useful exploration signature. However, anomalous values may also result from anisotropy.
The polar anisotropic version (See Thomsen, 1993, supra, and Rueger, “P-Wave Reflection Coefficients for Transversely Isotropic Media with Vertical and Horizontal Axis of Symmetry,” Soc. Expl. Geoph. Expdd. Absts. 1995, 65, 278-281, the teachings of which are incorporated herein by reference) of Equation (10) is:Rp(θ′)=R0′+R1′ sin 2θ′  (11) where: θ′ is the wave-front-normal angle;                                           R            0            ′                    =                                    Δ              ⁢                                                           ⁢                              Z                0                                                    2              ⁢                              Z                0                                                    ;        and                            (                  11          ⁢          a                )                                          R          1          ′                =                              1            2                    ⁢                      (                                                            Δ                  ⁢                                                                           ⁢                                      V                    0                                                                    V                  0                                            -                                                                    (                                                                  2                        ⁢                                                  V                          s0                                                                                            V                        0                                                              )                                    2                                ⁢                                                      Δ                    ⁢                                                                                   ⁢                                          G                      0                                                                            G                    0                                                              +                              Δ                ⁢                                                                   ⁢                δ                                      )                                              (                  11          ⁢          b                )            Here, Z0 is the vertical P-impedance, V0 is the vertical P-velocity, Vs0 is the vertical shear velocity, G0 is the vertical shear modulus, and Δ signifies a jump at the reflecting horizon. R1′ also contains the jump in anisotropy parameter δ across the reflecting horizon. The ray angle θ′ differs from the ray angle θ by an anisotropic correction factor involving δ (See Thomsen, 1986, supra).
It is clear from Equation (11) that, if the anisotropic contribution is present but ignored, then an anomalous value of R1′, due to gas may be masked (i.e., a false negative) or mimicked (i.e., false positive) by the anisotropy. Therefore, in order to interpret R1′, in terms of gas, the anisotropic contribution must first be estimated and corrected for. However, this estimation must be locally determined, with a resolution of ¼ wavelength or less; hence, the process of Equations 8 and 9, offering resolution of more than one wavelength, is not suitable.
Thus, a practical method and apparatus to do this with suitable resolution is needed.